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## G = C5×C22.46C24order 320 = 26·5

### Direct product of C5 and C22.46C24

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C22.46C24
 Chief series C1 — C2 — C22 — C2×C10 — C2×C20 — C5×C4⋊C4 — C5×C42.C2 — C5×C22.46C24
 Lower central C1 — C22 — C5×C22.46C24
 Upper central C1 — C2×C10 — C5×C22.46C24

Generators and relations for C5×C22.46C24
G = < a,b,c,d,e,f,g | a5=b2=c2=g2=1, d2=f2=b, e2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 282 in 214 conjugacy classes, 150 normal (62 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42.C2, C422C2, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22.46C24, C4×C20, C4×C20, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×C20, C22×C20, D4×C10, Q8×C10, C10×C4⋊C4, C5×C42⋊C2, C5×C42⋊C2, D4×C20, Q8×C20, C5×C22⋊Q8, C5×C22.D4, C5×C42.C2, C5×C42.C2, C5×C422C2, C5×C22.46C24
Quotients: C1, C2, C22, C5, C23, C10, C4○D4, C24, C2×C10, C2×C4○D4, 2- 1+4, C22×C10, C22.46C24, C5×C4○D4, C23×C10, C10×C4○D4, C5×2- 1+4, C5×C22.46C24

Smallest permutation representation of C5×C22.46C24
On 160 points
Generators in S160
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)(81 82 83 84 85)(86 87 88 89 90)(91 92 93 94 95)(96 97 98 99 100)(101 102 103 104 105)(106 107 108 109 110)(111 112 113 114 115)(116 117 118 119 120)(121 122 123 124 125)(126 127 128 129 130)(131 132 133 134 135)(136 137 138 139 140)(141 142 143 144 145)(146 147 148 149 150)(151 152 153 154 155)(156 157 158 159 160)
(1 35)(2 31)(3 32)(4 33)(5 34)(6 160)(7 156)(8 157)(9 158)(10 159)(11 17)(12 18)(13 19)(14 20)(15 16)(21 27)(22 28)(23 29)(24 30)(25 26)(36 55)(37 51)(38 52)(39 53)(40 54)(41 47)(42 48)(43 49)(44 50)(45 46)(56 75)(57 71)(58 72)(59 73)(60 74)(61 67)(62 68)(63 69)(64 70)(65 66)(76 95)(77 91)(78 92)(79 93)(80 94)(81 87)(82 88)(83 89)(84 90)(85 86)(96 115)(97 111)(98 112)(99 113)(100 114)(101 107)(102 108)(103 109)(104 110)(105 106)(116 135)(117 131)(118 132)(119 133)(120 134)(121 127)(122 128)(123 129)(124 130)(125 126)(136 155)(137 151)(138 152)(139 153)(140 154)(141 147)(142 148)(143 149)(144 150)(145 146)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 20)(7 16)(8 17)(9 18)(10 19)(11 157)(12 158)(13 159)(14 160)(15 156)(26 34)(27 35)(28 31)(29 32)(30 33)(36 41)(37 42)(38 43)(39 44)(40 45)(46 54)(47 55)(48 51)(49 52)(50 53)(56 61)(57 62)(58 63)(59 64)(60 65)(66 74)(67 75)(68 71)(69 72)(70 73)(76 81)(77 82)(78 83)(79 84)(80 85)(86 94)(87 95)(88 91)(89 92)(90 93)(96 101)(97 102)(98 103)(99 104)(100 105)(106 114)(107 115)(108 111)(109 112)(110 113)(116 121)(117 122)(118 123)(119 124)(120 125)(126 134)(127 135)(128 131)(129 132)(130 133)(136 141)(137 142)(138 143)(139 144)(140 145)(146 154)(147 155)(148 151)(149 152)(150 153)
(1 156 35 7)(2 157 31 8)(3 158 32 9)(4 159 33 10)(5 160 34 6)(11 28 17 22)(12 29 18 23)(13 30 19 24)(14 26 20 25)(15 27 16 21)(36 155 55 136)(37 151 51 137)(38 152 52 138)(39 153 53 139)(40 154 54 140)(41 147 47 141)(42 148 48 142)(43 149 49 143)(44 150 50 144)(45 146 46 145)(56 127 75 121)(57 128 71 122)(58 129 72 123)(59 130 73 124)(60 126 74 125)(61 135 67 116)(62 131 68 117)(63 132 69 118)(64 133 70 119)(65 134 66 120)(76 101 95 107)(77 102 91 108)(78 103 92 109)(79 104 93 110)(80 105 94 106)(81 96 87 115)(82 97 88 111)(83 98 89 112)(84 99 90 113)(85 100 86 114)
(1 101 21 96)(2 102 22 97)(3 103 23 98)(4 104 24 99)(5 105 25 100)(6 94 20 86)(7 95 16 87)(8 91 17 88)(9 92 18 89)(10 93 19 90)(11 82 157 77)(12 83 158 78)(13 84 159 79)(14 85 160 80)(15 81 156 76)(26 114 34 106)(27 115 35 107)(28 111 31 108)(29 112 32 109)(30 113 33 110)(36 121 41 116)(37 122 42 117)(38 123 43 118)(39 124 44 119)(40 125 45 120)(46 134 54 126)(47 135 55 127)(48 131 51 128)(49 132 52 129)(50 133 53 130)(56 141 61 136)(57 142 62 137)(58 143 63 138)(59 144 64 139)(60 145 65 140)(66 154 74 146)(67 155 75 147)(68 151 71 148)(69 152 72 149)(70 153 73 150)
(1 75 35 56)(2 71 31 57)(3 72 32 58)(4 73 33 59)(5 74 34 60)(6 134 160 120)(7 135 156 116)(8 131 157 117)(9 132 158 118)(10 133 159 119)(11 122 17 128)(12 123 18 129)(13 124 19 130)(14 125 20 126)(15 121 16 127)(21 67 27 61)(22 68 28 62)(23 69 29 63)(24 70 30 64)(25 66 26 65)(36 95 55 76)(37 91 51 77)(38 92 52 78)(39 93 53 79)(40 94 54 80)(41 87 47 81)(42 88 48 82)(43 89 49 83)(44 90 50 84)(45 86 46 85)(96 155 115 136)(97 151 111 137)(98 152 112 138)(99 153 113 139)(100 154 114 140)(101 147 107 141)(102 148 108 142)(103 149 109 143)(104 150 110 144)(105 146 106 145)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 140)(7 136)(8 137)(9 138)(10 139)(11 148)(12 149)(13 150)(14 146)(15 147)(16 141)(17 142)(18 143)(19 144)(20 145)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(56 76)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)(73 93)(74 94)(75 95)(96 135)(97 131)(98 132)(99 133)(100 134)(101 127)(102 128)(103 129)(104 130)(105 126)(106 125)(107 121)(108 122)(109 123)(110 124)(111 117)(112 118)(113 119)(114 120)(115 116)(151 157)(152 158)(153 159)(154 160)(155 156)

G:=sub<Sym(160)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,35)(2,31)(3,32)(4,33)(5,34)(6,160)(7,156)(8,157)(9,158)(10,159)(11,17)(12,18)(13,19)(14,20)(15,16)(21,27)(22,28)(23,29)(24,30)(25,26)(36,55)(37,51)(38,52)(39,53)(40,54)(41,47)(42,48)(43,49)(44,50)(45,46)(56,75)(57,71)(58,72)(59,73)(60,74)(61,67)(62,68)(63,69)(64,70)(65,66)(76,95)(77,91)(78,92)(79,93)(80,94)(81,87)(82,88)(83,89)(84,90)(85,86)(96,115)(97,111)(98,112)(99,113)(100,114)(101,107)(102,108)(103,109)(104,110)(105,106)(116,135)(117,131)(118,132)(119,133)(120,134)(121,127)(122,128)(123,129)(124,130)(125,126)(136,155)(137,151)(138,152)(139,153)(140,154)(141,147)(142,148)(143,149)(144,150)(145,146), (1,21)(2,22)(3,23)(4,24)(5,25)(6,20)(7,16)(8,17)(9,18)(10,19)(11,157)(12,158)(13,159)(14,160)(15,156)(26,34)(27,35)(28,31)(29,32)(30,33)(36,41)(37,42)(38,43)(39,44)(40,45)(46,54)(47,55)(48,51)(49,52)(50,53)(56,61)(57,62)(58,63)(59,64)(60,65)(66,74)(67,75)(68,71)(69,72)(70,73)(76,81)(77,82)(78,83)(79,84)(80,85)(86,94)(87,95)(88,91)(89,92)(90,93)(96,101)(97,102)(98,103)(99,104)(100,105)(106,114)(107,115)(108,111)(109,112)(110,113)(116,121)(117,122)(118,123)(119,124)(120,125)(126,134)(127,135)(128,131)(129,132)(130,133)(136,141)(137,142)(138,143)(139,144)(140,145)(146,154)(147,155)(148,151)(149,152)(150,153), (1,156,35,7)(2,157,31,8)(3,158,32,9)(4,159,33,10)(5,160,34,6)(11,28,17,22)(12,29,18,23)(13,30,19,24)(14,26,20,25)(15,27,16,21)(36,155,55,136)(37,151,51,137)(38,152,52,138)(39,153,53,139)(40,154,54,140)(41,147,47,141)(42,148,48,142)(43,149,49,143)(44,150,50,144)(45,146,46,145)(56,127,75,121)(57,128,71,122)(58,129,72,123)(59,130,73,124)(60,126,74,125)(61,135,67,116)(62,131,68,117)(63,132,69,118)(64,133,70,119)(65,134,66,120)(76,101,95,107)(77,102,91,108)(78,103,92,109)(79,104,93,110)(80,105,94,106)(81,96,87,115)(82,97,88,111)(83,98,89,112)(84,99,90,113)(85,100,86,114), (1,101,21,96)(2,102,22,97)(3,103,23,98)(4,104,24,99)(5,105,25,100)(6,94,20,86)(7,95,16,87)(8,91,17,88)(9,92,18,89)(10,93,19,90)(11,82,157,77)(12,83,158,78)(13,84,159,79)(14,85,160,80)(15,81,156,76)(26,114,34,106)(27,115,35,107)(28,111,31,108)(29,112,32,109)(30,113,33,110)(36,121,41,116)(37,122,42,117)(38,123,43,118)(39,124,44,119)(40,125,45,120)(46,134,54,126)(47,135,55,127)(48,131,51,128)(49,132,52,129)(50,133,53,130)(56,141,61,136)(57,142,62,137)(58,143,63,138)(59,144,64,139)(60,145,65,140)(66,154,74,146)(67,155,75,147)(68,151,71,148)(69,152,72,149)(70,153,73,150), (1,75,35,56)(2,71,31,57)(3,72,32,58)(4,73,33,59)(5,74,34,60)(6,134,160,120)(7,135,156,116)(8,131,157,117)(9,132,158,118)(10,133,159,119)(11,122,17,128)(12,123,18,129)(13,124,19,130)(14,125,20,126)(15,121,16,127)(21,67,27,61)(22,68,28,62)(23,69,29,63)(24,70,30,64)(25,66,26,65)(36,95,55,76)(37,91,51,77)(38,92,52,78)(39,93,53,79)(40,94,54,80)(41,87,47,81)(42,88,48,82)(43,89,49,83)(44,90,50,84)(45,86,46,85)(96,155,115,136)(97,151,111,137)(98,152,112,138)(99,153,113,139)(100,154,114,140)(101,147,107,141)(102,148,108,142)(103,149,109,143)(104,150,110,144)(105,146,106,145), (1,36)(2,37)(3,38)(4,39)(5,40)(6,140)(7,136)(8,137)(9,138)(10,139)(11,148)(12,149)(13,150)(14,146)(15,147)(16,141)(17,142)(18,143)(19,144)(20,145)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(56,76)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(96,135)(97,131)(98,132)(99,133)(100,134)(101,127)(102,128)(103,129)(104,130)(105,126)(106,125)(107,121)(108,122)(109,123)(110,124)(111,117)(112,118)(113,119)(114,120)(115,116)(151,157)(152,158)(153,159)(154,160)(155,156)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,35)(2,31)(3,32)(4,33)(5,34)(6,160)(7,156)(8,157)(9,158)(10,159)(11,17)(12,18)(13,19)(14,20)(15,16)(21,27)(22,28)(23,29)(24,30)(25,26)(36,55)(37,51)(38,52)(39,53)(40,54)(41,47)(42,48)(43,49)(44,50)(45,46)(56,75)(57,71)(58,72)(59,73)(60,74)(61,67)(62,68)(63,69)(64,70)(65,66)(76,95)(77,91)(78,92)(79,93)(80,94)(81,87)(82,88)(83,89)(84,90)(85,86)(96,115)(97,111)(98,112)(99,113)(100,114)(101,107)(102,108)(103,109)(104,110)(105,106)(116,135)(117,131)(118,132)(119,133)(120,134)(121,127)(122,128)(123,129)(124,130)(125,126)(136,155)(137,151)(138,152)(139,153)(140,154)(141,147)(142,148)(143,149)(144,150)(145,146), (1,21)(2,22)(3,23)(4,24)(5,25)(6,20)(7,16)(8,17)(9,18)(10,19)(11,157)(12,158)(13,159)(14,160)(15,156)(26,34)(27,35)(28,31)(29,32)(30,33)(36,41)(37,42)(38,43)(39,44)(40,45)(46,54)(47,55)(48,51)(49,52)(50,53)(56,61)(57,62)(58,63)(59,64)(60,65)(66,74)(67,75)(68,71)(69,72)(70,73)(76,81)(77,82)(78,83)(79,84)(80,85)(86,94)(87,95)(88,91)(89,92)(90,93)(96,101)(97,102)(98,103)(99,104)(100,105)(106,114)(107,115)(108,111)(109,112)(110,113)(116,121)(117,122)(118,123)(119,124)(120,125)(126,134)(127,135)(128,131)(129,132)(130,133)(136,141)(137,142)(138,143)(139,144)(140,145)(146,154)(147,155)(148,151)(149,152)(150,153), (1,156,35,7)(2,157,31,8)(3,158,32,9)(4,159,33,10)(5,160,34,6)(11,28,17,22)(12,29,18,23)(13,30,19,24)(14,26,20,25)(15,27,16,21)(36,155,55,136)(37,151,51,137)(38,152,52,138)(39,153,53,139)(40,154,54,140)(41,147,47,141)(42,148,48,142)(43,149,49,143)(44,150,50,144)(45,146,46,145)(56,127,75,121)(57,128,71,122)(58,129,72,123)(59,130,73,124)(60,126,74,125)(61,135,67,116)(62,131,68,117)(63,132,69,118)(64,133,70,119)(65,134,66,120)(76,101,95,107)(77,102,91,108)(78,103,92,109)(79,104,93,110)(80,105,94,106)(81,96,87,115)(82,97,88,111)(83,98,89,112)(84,99,90,113)(85,100,86,114), (1,101,21,96)(2,102,22,97)(3,103,23,98)(4,104,24,99)(5,105,25,100)(6,94,20,86)(7,95,16,87)(8,91,17,88)(9,92,18,89)(10,93,19,90)(11,82,157,77)(12,83,158,78)(13,84,159,79)(14,85,160,80)(15,81,156,76)(26,114,34,106)(27,115,35,107)(28,111,31,108)(29,112,32,109)(30,113,33,110)(36,121,41,116)(37,122,42,117)(38,123,43,118)(39,124,44,119)(40,125,45,120)(46,134,54,126)(47,135,55,127)(48,131,51,128)(49,132,52,129)(50,133,53,130)(56,141,61,136)(57,142,62,137)(58,143,63,138)(59,144,64,139)(60,145,65,140)(66,154,74,146)(67,155,75,147)(68,151,71,148)(69,152,72,149)(70,153,73,150), (1,75,35,56)(2,71,31,57)(3,72,32,58)(4,73,33,59)(5,74,34,60)(6,134,160,120)(7,135,156,116)(8,131,157,117)(9,132,158,118)(10,133,159,119)(11,122,17,128)(12,123,18,129)(13,124,19,130)(14,125,20,126)(15,121,16,127)(21,67,27,61)(22,68,28,62)(23,69,29,63)(24,70,30,64)(25,66,26,65)(36,95,55,76)(37,91,51,77)(38,92,52,78)(39,93,53,79)(40,94,54,80)(41,87,47,81)(42,88,48,82)(43,89,49,83)(44,90,50,84)(45,86,46,85)(96,155,115,136)(97,151,111,137)(98,152,112,138)(99,153,113,139)(100,154,114,140)(101,147,107,141)(102,148,108,142)(103,149,109,143)(104,150,110,144)(105,146,106,145), (1,36)(2,37)(3,38)(4,39)(5,40)(6,140)(7,136)(8,137)(9,138)(10,139)(11,148)(12,149)(13,150)(14,146)(15,147)(16,141)(17,142)(18,143)(19,144)(20,145)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(56,76)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(96,135)(97,131)(98,132)(99,133)(100,134)(101,127)(102,128)(103,129)(104,130)(105,126)(106,125)(107,121)(108,122)(109,123)(110,124)(111,117)(112,118)(113,119)(114,120)(115,116)(151,157)(152,158)(153,159)(154,160)(155,156) );

G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80),(81,82,83,84,85),(86,87,88,89,90),(91,92,93,94,95),(96,97,98,99,100),(101,102,103,104,105),(106,107,108,109,110),(111,112,113,114,115),(116,117,118,119,120),(121,122,123,124,125),(126,127,128,129,130),(131,132,133,134,135),(136,137,138,139,140),(141,142,143,144,145),(146,147,148,149,150),(151,152,153,154,155),(156,157,158,159,160)], [(1,35),(2,31),(3,32),(4,33),(5,34),(6,160),(7,156),(8,157),(9,158),(10,159),(11,17),(12,18),(13,19),(14,20),(15,16),(21,27),(22,28),(23,29),(24,30),(25,26),(36,55),(37,51),(38,52),(39,53),(40,54),(41,47),(42,48),(43,49),(44,50),(45,46),(56,75),(57,71),(58,72),(59,73),(60,74),(61,67),(62,68),(63,69),(64,70),(65,66),(76,95),(77,91),(78,92),(79,93),(80,94),(81,87),(82,88),(83,89),(84,90),(85,86),(96,115),(97,111),(98,112),(99,113),(100,114),(101,107),(102,108),(103,109),(104,110),(105,106),(116,135),(117,131),(118,132),(119,133),(120,134),(121,127),(122,128),(123,129),(124,130),(125,126),(136,155),(137,151),(138,152),(139,153),(140,154),(141,147),(142,148),(143,149),(144,150),(145,146)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,20),(7,16),(8,17),(9,18),(10,19),(11,157),(12,158),(13,159),(14,160),(15,156),(26,34),(27,35),(28,31),(29,32),(30,33),(36,41),(37,42),(38,43),(39,44),(40,45),(46,54),(47,55),(48,51),(49,52),(50,53),(56,61),(57,62),(58,63),(59,64),(60,65),(66,74),(67,75),(68,71),(69,72),(70,73),(76,81),(77,82),(78,83),(79,84),(80,85),(86,94),(87,95),(88,91),(89,92),(90,93),(96,101),(97,102),(98,103),(99,104),(100,105),(106,114),(107,115),(108,111),(109,112),(110,113),(116,121),(117,122),(118,123),(119,124),(120,125),(126,134),(127,135),(128,131),(129,132),(130,133),(136,141),(137,142),(138,143),(139,144),(140,145),(146,154),(147,155),(148,151),(149,152),(150,153)], [(1,156,35,7),(2,157,31,8),(3,158,32,9),(4,159,33,10),(5,160,34,6),(11,28,17,22),(12,29,18,23),(13,30,19,24),(14,26,20,25),(15,27,16,21),(36,155,55,136),(37,151,51,137),(38,152,52,138),(39,153,53,139),(40,154,54,140),(41,147,47,141),(42,148,48,142),(43,149,49,143),(44,150,50,144),(45,146,46,145),(56,127,75,121),(57,128,71,122),(58,129,72,123),(59,130,73,124),(60,126,74,125),(61,135,67,116),(62,131,68,117),(63,132,69,118),(64,133,70,119),(65,134,66,120),(76,101,95,107),(77,102,91,108),(78,103,92,109),(79,104,93,110),(80,105,94,106),(81,96,87,115),(82,97,88,111),(83,98,89,112),(84,99,90,113),(85,100,86,114)], [(1,101,21,96),(2,102,22,97),(3,103,23,98),(4,104,24,99),(5,105,25,100),(6,94,20,86),(7,95,16,87),(8,91,17,88),(9,92,18,89),(10,93,19,90),(11,82,157,77),(12,83,158,78),(13,84,159,79),(14,85,160,80),(15,81,156,76),(26,114,34,106),(27,115,35,107),(28,111,31,108),(29,112,32,109),(30,113,33,110),(36,121,41,116),(37,122,42,117),(38,123,43,118),(39,124,44,119),(40,125,45,120),(46,134,54,126),(47,135,55,127),(48,131,51,128),(49,132,52,129),(50,133,53,130),(56,141,61,136),(57,142,62,137),(58,143,63,138),(59,144,64,139),(60,145,65,140),(66,154,74,146),(67,155,75,147),(68,151,71,148),(69,152,72,149),(70,153,73,150)], [(1,75,35,56),(2,71,31,57),(3,72,32,58),(4,73,33,59),(5,74,34,60),(6,134,160,120),(7,135,156,116),(8,131,157,117),(9,132,158,118),(10,133,159,119),(11,122,17,128),(12,123,18,129),(13,124,19,130),(14,125,20,126),(15,121,16,127),(21,67,27,61),(22,68,28,62),(23,69,29,63),(24,70,30,64),(25,66,26,65),(36,95,55,76),(37,91,51,77),(38,92,52,78),(39,93,53,79),(40,94,54,80),(41,87,47,81),(42,88,48,82),(43,89,49,83),(44,90,50,84),(45,86,46,85),(96,155,115,136),(97,151,111,137),(98,152,112,138),(99,153,113,139),(100,154,114,140),(101,147,107,141),(102,148,108,142),(103,149,109,143),(104,150,110,144),(105,146,106,145)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,140),(7,136),(8,137),(9,138),(10,139),(11,148),(12,149),(13,150),(14,146),(15,147),(16,141),(17,142),(18,143),(19,144),(20,145),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,55),(56,76),(57,77),(58,78),(59,79),(60,80),(61,81),(62,82),(63,83),(64,84),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(71,91),(72,92),(73,93),(74,94),(75,95),(96,135),(97,131),(98,132),(99,133),(100,134),(101,127),(102,128),(103,129),(104,130),(105,126),(106,125),(107,121),(108,122),(109,123),(110,124),(111,117),(112,118),(113,119),(114,120),(115,116),(151,157),(152,158),(153,159),(154,160),(155,156)]])

125 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A ··· 4J 4K ··· 4R 5A 5B 5C 5D 10A ··· 10L 10M ··· 10T 10U 10V 10W 10X 20A ··· 20AN 20AO ··· 20BT order 1 2 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 4 2 ··· 2 4 ··· 4 1 1 1 1 1 ··· 1 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4

125 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 C10 C10 C4○D4 C4○D4 C5×C4○D4 C5×C4○D4 2- 1+4 C5×2- 1+4 kernel C5×C22.46C24 C10×C4⋊C4 C5×C42⋊C2 D4×C20 Q8×C20 C5×C22⋊Q8 C5×C22.D4 C5×C42.C2 C5×C42⋊2C2 C22.46C24 C2×C4⋊C4 C42⋊C2 C4×D4 C4×Q8 C22⋊Q8 C22.D4 C42.C2 C42⋊2C2 C20 C2×C10 C4 C22 C10 C2 # reps 1 1 3 1 1 2 2 3 2 4 4 12 4 4 8 8 12 8 4 4 16 16 1 4

Matrix representation of C5×C22.46C24 in GL4(𝔽41) generated by

 10 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 0 9 0 0 32 0 0 0 0 0 36 23 0 0 6 5
,
 9 0 0 0 0 9 0 0 0 0 1 0 0 0 4 40
,
 0 1 0 0 1 0 0 0 0 0 9 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 4 39 0 0 28 37
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[0,32,0,0,9,0,0,0,0,0,36,6,0,0,23,5],[9,0,0,0,0,9,0,0,0,0,1,4,0,0,0,40],[0,1,0,0,1,0,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,4,28,0,0,39,37] >;

C5×C22.46C24 in GAP, Magma, Sage, TeX

C_5\times C_2^2._{46}C_2^4
% in TeX

G:=Group("C5xC2^2.46C2^4");
// GroupNames label

G:=SmallGroup(320,1554);
// by ID

G=gap.SmallGroup(320,1554);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,1128,3446,436,1242]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=g^2=1,d^2=f^2=b,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e^-1=b*d=d*b,g*e*g=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
// generators/relations

׿
×
𝔽